mulclpr - Intuitionistic Logic Explorer

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Theorem mulclpr 7570

Description: Closure of multiplication on positive reals. First statement of Proposition 9-3.7 of [Gleason] p. 124. (Contributed by NM, 13-Mar-1996.)

**Assertion**
Ref	Expression
mulclpr	⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 ·_P 𝐵) ∈ P)

Proof of Theorem mulclpr Dummy variables 𝑞 𝑟 𝑡 𝑢 𝑣 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-imp 7467	. . . 4 ⊢ ·_P = (𝑤 ∈ P, 𝑣 ∈ P ↦ ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (1^st ‘𝑤) ∧ 𝑧 ∈ (1^st ‘𝑣) ∧ 𝑥 = (𝑦 ·_Q 𝑧))}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (2^nd ‘𝑤) ∧ 𝑧 ∈ (2^nd ‘𝑣) ∧ 𝑥 = (𝑦 ·_Q 𝑧))}⟩)
2	1	genpelxp 7509	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 ·_P 𝐵) ∈ (𝒫 Q × 𝒫 Q))
3		mulclnq 7374	. . . 4 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦 ·_Q 𝑧) ∈ Q)
4	1, 3	genpml 7515	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ∃𝑞 ∈ Q 𝑞 ∈ (1^st ‘(𝐴 ·_P 𝐵)))
5	1, 3	genpmu 7516	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ∃𝑟 ∈ Q 𝑟 ∈ (2^nd ‘(𝐴 ·_P 𝐵)))
6	2, 4, 5	jca32 310	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ((𝐴 ·_P 𝐵) ∈ (𝒫 Q × 𝒫 Q) ∧ (∃𝑞 ∈ Q 𝑞 ∈ (1^st ‘(𝐴 ·_P 𝐵)) ∧ ∃𝑟 ∈ Q 𝑟 ∈ (2^nd ‘(𝐴 ·_P 𝐵)))))
7		ltmnqg 7399	. . . . 5 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑥 <_Q 𝑦 ↔ (𝑧 ·_Q 𝑥) <_Q (𝑧 ·_Q 𝑦)))
8		mulcomnqg 7381	. . . . 5 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (𝑥 ·_Q 𝑦) = (𝑦 ·_Q 𝑥))
9		mulnqprl 7566	. . . . 5 ⊢ ((((𝐴 ∈ P ∧ 𝑢 ∈ (1^st ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝑡 ∈ (1^st ‘𝐵))) ∧ 𝑥 ∈ Q) → (𝑥 <_Q (𝑢 ·_Q 𝑡) → 𝑥 ∈ (1^st ‘(𝐴 ·_P 𝐵))))
10	1, 3, 7, 8, 9	genprndl 7519	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ∀𝑞 ∈ Q (𝑞 ∈ (1^st ‘(𝐴 ·_P 𝐵)) ↔ ∃𝑟 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘(𝐴 ·_P 𝐵)))))
11		mulnqpru 7567	. . . . 5 ⊢ ((((𝐴 ∈ P ∧ 𝑢 ∈ (2^nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝑡 ∈ (2^nd ‘𝐵))) ∧ 𝑥 ∈ Q) → ((𝑢 ·_Q 𝑡) <_Q 𝑥 → 𝑥 ∈ (2^nd ‘(𝐴 ·_P 𝐵))))
12	1, 3, 7, 8, 11	genprndu 7520	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ∀𝑟 ∈ Q (𝑟 ∈ (2^nd ‘(𝐴 ·_P 𝐵)) ↔ ∃𝑞 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑞 ∈ (2^nd ‘(𝐴 ·_P 𝐵)))))
13	10, 12	jca 306	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (∀𝑞 ∈ Q (𝑞 ∈ (1^st ‘(𝐴 ·_P 𝐵)) ↔ ∃𝑟 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘(𝐴 ·_P 𝐵)))) ∧ ∀𝑟 ∈ Q (𝑟 ∈ (2^nd ‘(𝐴 ·_P 𝐵)) ↔ ∃𝑞 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑞 ∈ (2^nd ‘(𝐴 ·_P 𝐵))))))
14	1, 3, 7, 8	genpdisj 7521	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ∀𝑞 ∈ Q ¬ (𝑞 ∈ (1^st ‘(𝐴 ·_P 𝐵)) ∧ 𝑞 ∈ (2^nd ‘(𝐴 ·_P 𝐵))))
15		mullocpr 7569	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <_Q 𝑟 → (𝑞 ∈ (1^st ‘(𝐴 ·_P 𝐵)) ∨ 𝑟 ∈ (2^nd ‘(𝐴 ·_P 𝐵)))))
16	13, 14, 15	3jca 1177	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ((∀𝑞 ∈ Q (𝑞 ∈ (1^st ‘(𝐴 ·_P 𝐵)) ↔ ∃𝑟 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘(𝐴 ·_P 𝐵)))) ∧ ∀𝑟 ∈ Q (𝑟 ∈ (2^nd ‘(𝐴 ·_P 𝐵)) ↔ ∃𝑞 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑞 ∈ (2^nd ‘(𝐴 ·_P 𝐵))))) ∧ ∀𝑞 ∈ Q ¬ (𝑞 ∈ (1^st ‘(𝐴 ·_P 𝐵)) ∧ 𝑞 ∈ (2^nd ‘(𝐴 ·_P 𝐵))) ∧ ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <_Q 𝑟 → (𝑞 ∈ (1^st ‘(𝐴 ·_P 𝐵)) ∨ 𝑟 ∈ (2^nd ‘(𝐴 ·_P 𝐵))))))
17		elnp1st2nd 7474	. 2 ⊢ ((𝐴 ·_P 𝐵) ∈ P ↔ (((𝐴 ·_P 𝐵) ∈ (𝒫 Q × 𝒫 Q) ∧ (∃𝑞 ∈ Q 𝑞 ∈ (1^st ‘(𝐴 ·_P 𝐵)) ∧ ∃𝑟 ∈ Q 𝑟 ∈ (2^nd ‘(𝐴 ·_P 𝐵)))) ∧ ((∀𝑞 ∈ Q (𝑞 ∈ (1^st ‘(𝐴 ·_P 𝐵)) ↔ ∃𝑟 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘(𝐴 ·_P 𝐵)))) ∧ ∀𝑟 ∈ Q (𝑟 ∈ (2^nd ‘(𝐴 ·_P 𝐵)) ↔ ∃𝑞 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑞 ∈ (2^nd ‘(𝐴 ·_P 𝐵))))) ∧ ∀𝑞 ∈ Q ¬ (𝑞 ∈ (1^st ‘(𝐴 ·_P 𝐵)) ∧ 𝑞 ∈ (2^nd ‘(𝐴 ·_P 𝐵))) ∧ ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <_Q 𝑟 → (𝑞 ∈ (1^st ‘(𝐴 ·_P 𝐵)) ∨ 𝑟 ∈ (2^nd ‘(𝐴 ·_P 𝐵)))))))
18	6, 16, 17	sylanbrc 417	1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 ·_P 𝐵) ∈ P)

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 708 ∧ w3a 978 ∈ wcel 2148 ∀wral 2455 ∃wrex 2456 𝒫 cpw 3575 class class class wbr 4003 × cxp 4624 ‘cfv 5216 (class class class)co 5874 1^st c1st 6138 2^nd c2nd 6139 Qcnq 7278 ·_Q cmq 7281 <_Q cltq 7283 Pcnp 7289 ·_P cmp 7292

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4118 ax-sep 4121 ax-nul 4129 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-iinf 4587

This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-iun 3888 df-br 4004 df-opab 4065 df-mpt 4066 df-tr 4102 df-eprel 4289 df-id 4293 df-po 4296 df-iso 4297 df-iord 4366 df-on 4368 df-suc 4371 df-iom 4590 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-f1 5221 df-fo 5222 df-f1o 5223 df-fv 5224 df-ov 5877 df-oprab 5878 df-mpo 5879 df-1st 6140 df-2nd 6141 df-recs 6305 df-irdg 6370 df-1o 6416 df-2o 6417 df-oadd 6420 df-omul 6421 df-er 6534 df-ec 6536 df-qs 6540 df-ni 7302 df-pli 7303 df-mi 7304 df-lti 7305 df-plpq 7342 df-mpq 7343 df-enq 7345 df-nqqs 7346 df-plqqs 7347 df-mqqs 7348 df-1nqqs 7349 df-rq 7350 df-ltnqqs 7351 df-enq0 7422 df-nq0 7423 df-0nq0 7424 df-plq0 7425 df-mq0 7426 df-inp 7464 df-imp 7467

This theorem is referenced by: mulnqprlemfl 7573 mulnqprlemfu 7574 mulnqpr 7575 mulassprg 7579 distrlem1prl 7580 distrlem1pru 7581 distrlem4prl 7582 distrlem4pru 7583 distrlem5prl 7584 distrlem5pru 7585 distrprg 7586 1idpr 7590 recexprlemex 7635 ltmprr 7640 mulcmpblnrlemg 7738 mulcmpblnr 7739 mulclsr 7752 mulcomsrg 7755 mulasssrg 7756 distrsrg 7757 m1m1sr 7759 1idsr 7766 00sr 7767 recexgt0sr 7771 mulgt0sr 7776 mulextsr1lem 7778 mulextsr1 7779 recidpirq 7856

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